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Derivative of Logarithmic Functions Calculator

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Example

Solved Problems

Difficult Problems

1

Here, we show you a step-by-step solved example of derivative of logarithmic functions. This solution was automatically generated by our smart calculator:

$\int_1^2x\ln\left(x\right)dx$
2

We can solve the integral $\int x\ln\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{x}$
3

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=\ln\left(x\right)}\\ \displaystyle{du=\frac{1}{x}dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=xdx}\\ \displaystyle{\int dv=\int xdx}\end{matrix}$
5

Solve the integral to find $v$

$v=\int xdx$
6

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$\frac{1}{2}x^2$

Multiplying fractions $\frac{1}{x} \times \frac{1}{2}$

$\frac{1}{2}x^2\ln\left|x\right|-\int_{1}^{2}\frac{1}{2x}x^2dx$

Multiplying the fraction by $x^2$

$\frac{1}{2}x^2\ln\left|x\right|-\int_{1}^{2}\frac{x^2}{2x}dx$

Simplify the fraction $\frac{x^2}{2x}$ by $x$

$\frac{1}{2}x^2\ln\left|x\right|-\int_{1}^{2}\frac{x}{2}dx$
7

Now replace the values of $u$, $du$ and $v$ in the last formula

$\left[\frac{1}{2}x^2\ln\left|x\right|\right]_{1}^{2}-\int_{1}^{2}\frac{x}{2}dx$

Take the constant $\frac{1}{2}$ out of the integral

$-\frac{1}{2}\int_{1}^{2} xdx$

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$-\left[\frac{1}{2}\cdot \frac{1}{2}x^2\right]_{1}^{2}$

Simplify the expression

$-\left[\frac{1}{4}x^2\right]_{1}^{2}$

Evaluate the definite integral

$- \left(\frac{1}{4}\cdot 2^2- \left(\frac{1}{4}\right)\cdot 1^2\right)$

Simplify the expression

$-\frac{3}{4}$
8

The integral $-\int_{1}^{2}\frac{x}{2}dx$ results in: $-\frac{3}{4}$

$-\frac{3}{4}$
9

Gather the results of all integrals

$\left[\frac{1}{2}x^2\ln\left|x\right|\right]_{1}^{2}-\frac{3}{4}$
10

Evaluate the definite integral

$\frac{1}{2}\cdot 2^2\ln\left|2\right|-eval\left(\frac{1}{2}x^2\ln\left|x\right|,1\right)-\frac{3}{4}$

Calculate the power $2^2$

$\left(\frac{1}{2}\right)\cdot 4\ln\left|2\right|-eval\left(\frac{1}{2}x^2\ln\left|x\right|,1\right)-\frac{3}{4}$

Multiply the fraction and term in $\left(\frac{1}{2}\right)\cdot 4\ln\left|2\right|$

$2\ln\left|2\right|-eval\left(\frac{1}{2}x^2\ln\left|x\right|,1\right)-\frac{3}{4}$

Calculate the numerical value of the expression $\frac{1}{2}x^2\ln\left|x\right|$

$2\ln\left|2\right|- \left(\frac{1}{2}\right)\cdot 1^2\ln\left|1\right|-\frac{3}{4}$

$1$ raised to any expression (except infinity) is equal to $1$

$2\ln\left|2\right|- \left(\frac{1}{2}\right)\ln\left|1\right|-\frac{3}{4}$

Multiplying the fraction by $-1$

$2\ln\left|2\right|-\frac{1}{2}\ln\left|1\right|-\frac{3}{4}$

Calculating the natural logarithm of $1$

$2\ln\left|2\right|+\left(-\frac{1}{2}\right)\cdot 0-\frac{3}{4}$

Any expression multiplied by $0$ is equal to $0$

$2\ln\left|2\right|-\frac{3}{4}$
11

Simplify the expression

$2\ln\left(2\right)-\frac{3}{4}$

Final answer to the problem

$2\ln\left(2\right)-\frac{3}{4}$

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